## Abstract

For a set of n points in ℝ^{d}, and parameters k and ε, we present a data structure that answers (1 + ε)-approximate k nearest neighbor queries in logarithmic time. Surprisingly, the space used by the data-structure is Õ(n/k), that is, the space used is sub linear in the input size if k is sufficiently large. Our approach provides a novel way to summarize geometric data, such that meaningful proximity queries on the data can be carried out using this sketch. Using this we provide a sub linear space data-structure that can estimate the density of a point set under various measures, including: (i) sum of distances of k closest points to the query point, and (ii) sum of squared distances of k closest points to the query point. Our approach generalizes to other distance based estimation of densities of similar flavor.

Original language | English (US) |
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Article number | 6375321 |

Pages (from-to) | 430-439 |

Number of pages | 10 |

Journal | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

DOIs | |

State | Published - 2012 |

Event | 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012 - New Brunswick, NJ, United States Duration: Oct 20 2012 → Oct 23 2012 |

## ASJC Scopus subject areas

- Computer Science(all)